Theorem crnggrpd 20219

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20218	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20214	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  Grpcgrp 18900  CRingccrg 20206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-ov 7363  df-ring 20207  df-cring 20208

This theorem is referenced by:  fermltlchr  21519  psdmul  22142  psd1  22143  psdpw  22146  ply1chr  22281  ply1fermltlchr  22287  elrgspnsubrunlem1  33323  elrgspnsubrunlem2  33324  erlbr2d  33340  rlocaddval  33344  rloccring  33346  rloc0g  33347  rlocf1  33349  fracerl  33382  gsumind  33420  ressply1evls1  33640  evl1deg1  33651  evl1deg2  33652  evl1deg3  33653  ply1dg1rt  33655  vr1nz  33668  psrmonprod  33711  mplmonprod  33713  esplyfvn  33736  irngss  33847  extdgfialglem1  33852  irredminply  33876  algextdeglem4  33880  algextdeglem5  33881  aks6d1c1p3  42563  aks6d1c2lem4  42580  aks6d1c6lem2  42624  aks5lem2  42640  evlsbagval  43016  selvvvval  43032  evlselv  43034  selvadd  43035  prjcrv0  43080